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Millikan’s Oil-Drop
Experiment
Millikan’s measurement of the charge on the electron is one of the few truly crucial
experiments in physics and, at the same time, one whose simple directness serves as a
standard against which to compare others. Figure 3-4 shows a sketch of Millikan’s
apparatus. With no electric field, the downward force on an oil drop is mg and the
upward force is bv. The equation of motion is
mg bv m
dv
dt
3-10
where b is given by Stokes’ law:
b 6a
3-11
and where is the coefficient of viscosity of the fluid (air) and a is the radius of the
drop. The terminal velocity of the falling drop vf is
Atomizer
(+)
(–)
(–)
Light
source
(+)
Telescope
Fig. 3-4 Schematic diagram of the Millikan oil-drop apparatus. The drops are sprayed from
the atomizer and pick up a static charge, a few falling through the hole in the top plate. Their
fall due to gravity and their rise due to the electric field between the capacitor plates can be
observed with the telescope. From measurements of the rise and fall times, the electric charge
on a drop can be calculated. The charge on a drop could be changed by exposure to x rays
from a source (not shown) mounted opposite the light source.
(Continued)
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vf Buoyant force bv
mg
b
3-12
(see Figure 3-5). When an electric field is applied, the upward motion of a charge
qn is given by
Droplet e
qn mg bv m
Weight mg
Fig. 3-5 An oil droplet
carrying an ion of charge e
falling at terminal speed,
i.e., mg bv.
dv
dt
Thus the terminal velocity vr of the drop rising in the presence of the electric field is
vr qn mg
b
3-13
In this experiment, the terminal speeds were reached almost immediately, and the
drops drifted a distance L upward or downward at a constant speed. Solving Equations 3-12 and 3-13 for qn , we have
qn mg
mgTf 1
1
(vf vr) vf
Tf Tr
3-14
where Tf L/vf is the fall time and Tr L/vr is the rise time.
If any additional charge is picked up, the terminal velocity becomes vr , which is
related to the new charge qn by Equation 3-13:
vr qn mg
b
The amount of charge gained is thus
mg
(v vr)
vf r
mgTf 1
1
Tr
Tr
qn qn 3-15
The velocities vf , vt , and vr are determined by measuring the time taken to fall or rise
the distance L between the capacitor plates.
If we write qn ne and qn qn ne where n is the change in n, Equations
3-14 and 3-15 can be written
3-16
3-17
1 1
1
e
n Tf Tr
mgTf
and
1
e
1 1
n Tr Tr
mgTf
(Continued)
Millikan’s Oil-Drop Experiment
To obtain the value of e from the measured fall and rise times, one needs to know the
mass of the drop (or its radius, since the density is known). The radius is obtained
from Stokes’ law using Equation 3-12.
Notice that the right sides of Equations 3-16 and 3-17 are equal to the same constant, albeit an unknown one, since it contains the unknown e. The technique, then,
was to obtain a drop in the field of view and measure its fall time Tf (electric field
off) and its rise time Tr (electric field on) for the unknown number of charges n on
the drop. Then, for the same drop (hence, same mass m), n was changed by some
unknown amount n by exposing the drop to the x-ray source, thereby yielding a new
value for n; and Tf and Tr were measured. The number of charges on the drop was
changed again and the fall and rise times recorded. This process was repeated over
and over for as long as the drop could be held in view (or until the experimenter
became tired), often for several hours at a time. The value of e was then determined
by finding (basically by trial and error) the integer values of n and n that made the
left sides of Equations 3-16 and 3-17 equal to the same constant for all measurements using a given drop.
Millikan did experiments like these with thousands of drops, some of nonconducting oil, some of semiconductors like glycerine, and some of conductors like mercury.
In no case was a charge found that was a fractional part of e. This process, which you
will have the opportunity to work with in solving the problem below using actual data
from Millikan’s sixth drop, yielded a value of e of 1.591 10
19 C. This value was
accepted for about 20 years, until it was discovered that x-ray diffraction measurements
of NA gave values of e that differed from Millikan’s by about 0.4 percent. The discrepancy was traced to the value of the coefficient of viscosity used by Millikan, which
was too low. Improved measurements of gave a value about 0.5 percent higher, thus
changing the value of e resulting from the oil-drop experiment to 1.601 10
19 C, in
good agreement with the x-ray diffraction data. The modern “best” values of e and
other physical constants are published periodically by the International Council of
Scientific Unions. The currently accepted value of the electron charge is8
e 1.60217733 10
19C
3-18
with an uncertainty of 0.30 parts per million. Our needs in this book are rarely as
precise as this, so we will typically use e 1.602 10
19 C. Note that, while we
have been able to measure the value of the quantized electric charge, there is no hint
in any of the above as to why it has this value, nor do we know the answer to that
question now.
Hardly a matter of only historical interest, Millikan’s technique is currently
being used in an ongoing search for elementary particles with fractional electric
charge by M. Perl and co-workers.
P ROBLEM
The accompanying table shows a portion of the data collected by Millikan for drop number 6
in the oil-drop experiment. (a) Find the terminal fall velocity vf from the table using the mean
fall time and the fall distance (10.21 mm). (b) Use the density of oil 0.943 g/cm3 943 kg/ m3, the viscosity of air 1.824 10
5 N s/m2, and g 9.81 m/s2 to calculate
the radius a of the oil drop from Stokes’ law as expressed in Equation 3-12. (c) The correct
“trial value” of n is filled in in column 7. Determine the remaining correct values for n and
n, in columns 4 and 7, respectively. (d) Compute e from the data in the table.
(Continued)
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Rise and fall times of a single oil drop with calculated number
of elementary charges on drop
1
Tf
2
Tr
11.848
80.708
11.890
22.366
11.908
22.390
11.904
22.368
11.882
3
1/Tr 1/Tr
4
n
5
(1/n)(1/Tr 1/Tr )
6
1/Tf 1/Tr
7
n
8
(1/n)(1/Tr 1/Tf)
0.09655
18
0.005366
0.03234
6
0.005390
0.12887
24
0.005371
140.566
0.03751
7
0.005358
0.09138
17
0.005375
11.906
79.600
0.005348
1
0.005348
0.09673
18
0.005374
11.838
34.748
0.01616
3
0.005387
0.11289
21
0.005376
11.816
34.762